Quantum topology in the ultrastrong coupling regime

The coupling between two or more objects can generally be categorized as strong or weak. In cavity quantum electrodynamics for example, when the coupling strength is larger than the loss rate the coupling is termed strong, and otherwise it is dubbed weak. Ultrastrong coupling, where the interaction energy is of the same order of magnitude as the bare energies of the uncoupled objects, presents a new paradigm for quantum physics and beyond. As a consequence profound changes to well established phenomena occur, for instance the ground state in an ultrastrongly coupled system is not empty but hosts virtual excitations due to the existence of processes which do not conserve the total number of excitations. The implications of ultrastrong coupling for quantum topological systems, where the number of excitations are typically conserved, remain largely unknown despite the great utility of topological matter. Here we reveal how the delicate interplay between ultrastrong coupling and topological states manifests in a one-dimensional array. We study theoretically a dimerized chain of two-level systems within the ultrastrong coupling regime, where the combined saturation and counter-rotating terms in the Hamiltonian are shown to play pivotal roles in the rich, multi-excitation effective bandstructure. In particular, we uncover unusual topological edge states, we introduce a flavour of topological state which we call an anti-edge state, and we reveal the remarkable geometric-dependent renormalizations of the quantum vaccum. Taken together, our results provide a route map for experimentalists to characterize and explore a prototypical system in the emerging field of ultrastrong quantum topology.

In this Supplementary Information we provide an in-depth treatment of various models of coupled two-level systems. We build up to the full dimerized chain theory within the ultrastrong coupling regime, which is the subject of the main text. In particular, we provide analytical results describing dimers, trimers, and regular chains of coupled two-level systems beyond the rotating wave approximation. Let us consider a pair of coupled two-level systems (2LSs), as discussed at the start of the main text. The Hamiltonian readŝ

CONTENTS
where the bare transition frequency is ω 0 , and the coupling strength is J. Arranging by the number N of excitations in the system, the bare state basis is leading to a representation of Eq. (S1) with the 4 × 4 matrix A. Strong coupling In the strong coupling regime one may apply the rotating wave approximation (RWA) to Eq. (S1), such that the counterrotating (C-R) terms ∝ σ † 1 σ † 2 and ∝ σ 1 σ 2 are discarded. Then the matrix representation of Eq. (S3) collapses into the block diagonal form where the three contributions (corresponding to the 0, 1, and 2 excitation sectors) are The Hamiltonian H may be diagonalized by Bogoliubov transformation into where the four strong coupling regime eigenfrequencies ω n are given by The associated eigenstates |ψ n in the strong coupling regime read Notably, the completely unoccupied ground state |ψ 1 and the doubly-occupied |ψ 4 are both at energies independent of the coupling strength J. The intermediate, singly-occupied states |ψ 2 and |ψ 3 are hybridized in nature, and display a Rabi splitting of ω 3 − ω 2 = 2J. These features are plotted in Fig. S1 (a), which displays the eigenfrequencies ω n as a function of J.

B. Ultrastrong coupling
Within the ultrastrong coupling regime the C-R terms in Eq. (S1) are important, so that the governing matrix Hamiltonian is given by Eq. (S3). Since these C-R terms link the 0 and 2 excitation sectors, while leaving the 1 excitation sector unchanged from Eq. (S5), one can rewrite Eq. (S3) as Diagonalization of these matrices leads to the transformation of Eq. (S1) into [cf. Eq. (S6)] where we have used ||ψ n to distinguish the exact eigenstates from those only valid in the strong coupling regime. The eigenfrequencies ω n in the ultrastrong coupling regime read [cf. Eq. (S7)] where we have introduced the frequencyω while the associated eigenstates read [cf. Eq. (S8)] Clearly, upon entering ultrastrong coupling the extremities of the ladder ω 1 and ω 4 become renormalized from their bare values, which arises due to the mixture of the bare states |0, 0 and |1, 1 inside ||ψ 1 and ||ψ 4 . This is made possible by the nonnumber conserving processes described by the C-R terms. These energy ladder reconstructions are clearly visible in Fig. S2 (a), which displays the full eigenfrequencies ω n as a function of J.
The dependence of the weightings of the bare states comprising the eigenstates ||ψ n with the coupling strength J is represented in Fig. S3. In the figure white represents zero overlap, and increasingly dark red denotes increasingly large overlap. Noticeably, the intermediate states are unaffected by the magnitude of J, while the ground and highest energy states provide the fingerprints of ultrastrong coupling by breaking the conservation of the number of excitations.

C. Correlations
In terms of the HamiltonianĤ given by Eq. (S1), the quantum Liouville-von Neumann equation for the density matrix ρ reads Upon using the property O = Tr (Oρ), which is valid for any operator O, one may find the first and second moments of the system. We consider each case in turn. First moments. The first moments of the system may be found from the equation of motion where u collects the mean values of the operators, via and where the 8 × 8 dynamical matrix M is defined as Subject to the initial condition of σ 1 = 1 and σ 2 = 0 at t = 0, the solution of Eq. (S15) yields the results where the renormalized frequencyω 0 is defined in Eq. (S12). The product of Eq. (S19) with their complex conjugate expressions for σ † 1 and σ † 2 yields Eq. (4) in the main text.
Second moments. Similarly, the second moments may be determined from the equation of motion where the mean values of the second moments w, the effective drive term P, and the 6 × 6 dynamical matrix Q are defined via With the initial condition σ † 1 σ 1 = 1 and σ † 2 σ 2 = 0 at t = 0, the solution of Eq. (S20) yields The lack of equivalence between σ † n σ n and σ † n σ n is due to corrections to the prior result of σ † n σ n .
The latter quantity ρ Tp appears indirectly, via its eigenvalues λ n , in the definition of the negativity N N = n |λ n | − λ n 2 , which is a measure of quantum entanglement.

II. A TRIPLE OF COUPLED TWO-LEVEL SYSTEMS
To build up further intuition, let us now consider a triple of coupled 2LSs, as captured by the Hamiltonian Where N counts the number of excitations, the bare state basis may be defined via such that the 8 × 8 matrix representation of Eq. (S36) readily follows as

A. Strong coupling
Within the RWA, the number of excitations is conserved so that Eq. (S37) reduces to the block diagonal form which is composed of the submatrices describing the 0, 1, 2, and 3 excitation sectors, where The eigenfrequencies ω n are as follows and are plotted in Fig. S1 (b), where the differences to immediately smaller and immediately larger chains may be seen in the panels either side of panel (b).

B. Ultrastrong coupling
The full governing matrix of Eq. (S37) can be reshuffled into two matrices, one describing the connected zero and two excitation subspace, and the other describing the one and three excitation subspace, as follows which yield exact expressions for the eigenfrequencies ω n where we have introduced the angular parameter These eigenfrequencies ω n are plotted in Fig. S2 (b) as a function of J, with complementary results available in the panels either side for shorter and longer chains. The fidelities of the eigenstates ||ψ n as a function of J are shown in Fig. S4, with the ordering from the ground state ||ψ 1 to the fully-occupied state ||ψ 8 . The influence of ultrastrong coupling is demonstrated in the breakdown of the excitation number conserving sectors. The strong coupling eigenfrequencies ω n follow directly as

A. Strong coupling
In the RWA, some general properties can nevertheless be drawn. SinceĤ becomes number conserving after discarding the non-resonant terms, the resulting Hamiltonian may be split into chunks describing a different number of excitations N , as is drawn in Fig. S6 (a) for small chains of size N . The size of each chunk is given by the binomial coefficient N !/N !/(N − N )!, such that the structure follows Pascal's triangle, as is sketched in Fig. S6 (b). Now let us consider the energy ladder of a general chain of size N . The N = 0 excitation sector remains trivial, with an empty ground state at zero energy, as one would expect. The N = 1 excitation sector corresponds to the standard cosine result of a one-dimensional tight-binding array. At the other end of the energy ladder, the N = N excitation sector is associated with the highest possible eigenfrequency N ω 0 , corresponding to a wholly excited, saturated state. The penultimate rung of the ladder, the N = N − 1 excitation sector, is a mirror image of the N = 1 case, with holes (or the absence of excitations) playing the role of excitations [it is this mirroring quality which allows for the so-called anti-edge states for a dimerized chain to emerge, as discussed in the main text]. These aforementioned results can be briefly listed as . . .
Elsewhere, in the intermediate rungs of the energy ladder, general analytic results are harder to write down.

B. Ultrastrong coupling
The C-R terms break the number conservation enjoyed in strong coupling. As shown in Fig. S6 (a), now all even-numbered and all odd-numbered excitation sectors are linked. In Fig. S7 we compare the eigenfrequencies in the strong (thick yellow lines) and ultrastrong (thin red lines) coupling regimes, as a function of the coupling strength J, for shorter chains from N = 2 to N = 5.